Let f be a function so that (below). Which must be true?I. f is continuous at x=2II. f is differentiable at x=2III. The derivative of f is continuous at x=2(A) I (B) II (C) I and II (D) I and III (E) II and III

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neizhojen50q · Accepted Answer

(C) Explanation: Noting that a function       f   is differentiable at a point             x      0       if            lim              h        →        0                                      f                      (                          x              0                        +            h            )                          -                  f                      (                          x              0                        )                              h        =    L  the given information effectively is that       f   is differentiable at       2   and that       f    ′          (      2      )        =    5  .Now, looking at the statements:I: TrueA function&#039;s differentiability at a point implies its continuity at that point.II: TrueThe given information matches the definition of differentiability at       x    =    2  .III: FalseThe derivative of a function is not necessarily continuous, a classic example being             g              (        x        )              =          {                                                  x              2                                      sin                              (                                  1                  x                                )                                                                  if                                      x            ≠            0                                                0                                        if                                      x            =            0                              , which is differentiable at       0  , but whose derivative has a discontinuity at       0  .

Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I and II (D) I and III (E) II and III

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